Theorem isdivrngo 38446

Description: The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)

Assertion

Ref	Expression
isdivrngo	⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))

Proof of Theorem isdivrngo

Dummy variables 𝑔 ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 5101	. . . . 5 ⊢ (𝐺DivRingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ DivRingOps)
2		df-drngo 38445	. . . . . . 7 ⊢ DivRingOps = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ RingOps ∧ (𝑦 ↾ ((ran 𝑥 ∖ {(GId‘𝑥)}) × (ran 𝑥 ∖ {(GId‘𝑥)}))) ∈ GrpOp)}
3	2	relopabiv 5793	. . . . . 6 ⊢ Rel DivRingOps
4	3	brrelex1i 5703	. . . . 5 ⊢ (𝐺DivRingOps𝐻 → 𝐺 ∈ V)
5	1, 4	sylbir 237	. . . 4 ⊢ (⟨𝐺, 𝐻⟩ ∈ DivRingOps → 𝐺 ∈ V)
6	5	anim1i 624	. . 3 ⊢ ((⟨𝐺, 𝐻⟩ ∈ DivRingOps ∧ 𝐻 ∈ 𝐴) → (𝐺 ∈ V ∧ 𝐻 ∈ 𝐴))
7	6	ancoms 462	. 2 ⊢ ((𝐻 ∈ 𝐴 ∧ ⟨𝐺, 𝐻⟩ ∈ DivRingOps) → (𝐺 ∈ V ∧ 𝐻 ∈ 𝐴))
8		rngoablo2 38405	. . . . 5 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp)
9		elex 3475	. . . . 5 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ V)
10	8, 9	syl 17	. . . 4 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ V)
11	10	ad2antrl 738	. . 3 ⊢ ((𝐻 ∈ 𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → 𝐺 ∈ V)
12		simpl 486	. . 3 ⊢ ((𝐻 ∈ 𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → 𝐻 ∈ 𝐴)
13	11, 12	jca 519	. 2 ⊢ ((𝐻 ∈ 𝐴 ∧ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)) → (𝐺 ∈ V ∧ 𝐻 ∈ 𝐴))
14		df-drngo 38445	. . . 4 ⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
15	14	eleq2i 2854	. . 3 ⊢ (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ ⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)})
16		opeq1 4831	. . . . . 6 ⊢ (𝑔 = 𝐺 → ⟨𝑔, ℎ⟩ = ⟨𝐺, ℎ⟩)
17	16	eleq1d 2847	. . . . 5 ⊢ (𝑔 = 𝐺 → (⟨𝑔, ℎ⟩ ∈ RingOps ↔ ⟨𝐺, ℎ⟩ ∈ RingOps))
18		rneq 5912	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
19		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
20	19	sneqd 4594	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → {(GId‘𝑔)} = {(GId‘𝐺)})
21	18, 20	difeq12d 4081	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (ran 𝑔 ∖ {(GId‘𝑔)}) = (ran 𝐺 ∖ {(GId‘𝐺)}))
22	21	sqxpeqd 5679	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)})))
23	22	reseq2d 5965	. . . . . 6 ⊢ (𝑔 = 𝐺 → (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))))
24	23	eleq1d 2847	. . . . 5 ⊢ (𝑔 = 𝐺 → ((ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp ↔ (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
25	17, 24	anbi12d 641	. . . 4 ⊢ (𝑔 = 𝐺 → ((⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp) ↔ (⟨𝐺, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
26		opeq2 4832	. . . . . 6 ⊢ (ℎ = 𝐻 → ⟨𝐺, ℎ⟩ = ⟨𝐺, 𝐻⟩)
27	26	eleq1d 2847	. . . . 5 ⊢ (ℎ = 𝐻 → (⟨𝐺, ℎ⟩ ∈ RingOps ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps))
28		reseq1 5959	. . . . . 6 ⊢ (ℎ = 𝐻 → (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) = (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))))
29	28	eleq1d 2847	. . . . 5 ⊢ (ℎ = 𝐻 → ((ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp ↔ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
30	27, 29	anbi12d 641	. . . 4 ⊢ (ℎ = 𝐻 → ((⟨𝐺, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp) ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
31	25, 30	opelopabg 5509	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) → (⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)} ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
32	15, 31	bitrid 285	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
33	7, 13, 32	pm5.21nd 811	1 ⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ∖ cdif 3901  {csn 4582  ⟨cop 4588   class class class wbr 5100  {copab 5162   × cxp 5645  ran crn 5648   ↾ cres 5649  ‘cfv 6521  GrpOpcgr 30689  GIdcgi 30690  AbelOpcablo 30744  RingOpscrngo 38390  DivRingOpscdrng 38444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-1st 7970  df-2nd 7971  df-rngo 38391  df-drngo 38445

This theorem is referenced by:  zrdivrng  38449  isdrngo1  38452